{"id":1261,"date":"2016-05-16T08:00:09","date_gmt":"2016-05-16T12:00:09","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=1261"},"modified":"2016-05-16T08:12:51","modified_gmt":"2016-05-16T12:12:51","slug":"believing-in-mathematics","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2016\/05\/16\/believing-in-mathematics\/","title":{"rendered":"Believing in Mathematics"},"content":{"rendered":"<p><i><span style=\"font-weight: 400\">By Benjamin Braun, <a href=\"http:\/\/blogs.ams.org\/matheducation\/about-the-editors\/\" target=\"_blank\">Editor-in-Chief<\/a>, University of Kentucky<\/span><\/i><\/p>\n<p><span style=\"font-weight: 400\">In my experience, many students in K-12 and post-secondary mathematics courses believe that: <\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400\">all math problems have known answers,<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">failure and misunderstanding are absent from successful mathematics,<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">their instructor can always find answers to problems, and<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">regardless of what instructors say, students will be judged and\/or assessed based on whether or not they can obtain correct answers to problems they are given.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">As long as students believe in this mythology, it is hard to motivate them to develop quality mathematical practices. In an effort to undercut these misunderstandings and unproductive beliefs about the nature of mathematics, over the past several years I\u2019ve experimented with assignments and activities that purposefully range across the intellectual, behavioral, and emotional psychological domains. In this article, I provide a toolbox of activities for faculty interested in incorporating these or similar interventions in their courses.<\/span><\/p>\n<p><b>Psychological Domains<\/b><\/p>\n<p><span style=\"font-weight: 400\">A useful oversimplification frames the human psyche as a three-stranded model:<\/span><\/p>\n<p><a href=\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png\" rel=\"attachment wp-att-1262\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1262 aligncenter\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=300%2C300\" alt=\"psyche\" width=\"300\" height=\"300\" srcset=\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=300%2C300&amp;ssl=1 300w, https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=150%2C150&amp;ssl=1 150w, https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=32%2C32&amp;ssl=1 32w, https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=64%2C64&amp;ssl=1 64w, https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=96%2C96&amp;ssl=1 96w, https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?resize=128%2C128&amp;ssl=1 128w, https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2016\/05\/psyche.png?w=429&amp;ssl=1 429w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><!--more--><\/p>\n<p><span style=\"font-weight: 400\">The intellectual, or <\/span><i><span style=\"font-weight: 400\">cognitive<\/span><\/i><span style=\"font-weight: 400\">, domain regards knowledge and understanding of concepts. The behavioral, or <\/span><i><span style=\"font-weight: 400\">enactive<\/span><\/i><span style=\"font-weight: 400\">, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or <\/span><i><span style=\"font-weight: 400\">affective<\/span><\/i><span style=\"font-weight: 400\">, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning. In post-secondary mathematics courses, our classroom activities and assessments often focus primarily on intellectual knowledge and understanding, with emotional and behavioral aspects of learning addressed either implicitly or not at all. A partial antidote to this is found in the many <\/span><a href=\"http:\/\/blogs.ams.org\/matheducation\/category\/active-learning-in-mathematics-series-2015\"><span style=\"font-weight: 400\">active learning techniques<\/span><\/a><span style=\"font-weight: 400\"> being used in post-secondary mathematics courses, such as think-pair-share, \u201cclicker\u201d systems, one-minute papers, inquiry-based learning, and service learning, among others. A strength of active learning methods is that they challenge students\u2019 unhelpful beliefs and practices through public dialogue and activities. What active learning techniques might not <\/span><i><span style=\"font-weight: 400\">explicitly<\/span><\/i><span style=\"font-weight: 400\"> do is frame these discussions and activities within a broader context involving the nature of intelligence and the process of successful learning. <\/span><\/p>\n<p><span style=\"font-weight: 400\">A goal for my courses is to incorporate direct interventions that provide students with three things:<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">language that supports articulate reflection and discussion in the context of emotional and behavioral domains,<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">an environment in which such reflection and discussion arise naturally and effectively, and<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">a contemporary \u201cexternal source\u201d motivating this language and environment so that our discussion is not driven by the will of the instructor.<\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400\">The ways in which these interventions are realized in my classes will change over time, and I am willing to follow current educational trends if they are effective tools for my students. Many of the interventions I have used are based on research in psychology regarding mindsets, a topic that <\/span><a href=\"http:\/\/blogs.ams.org\/matheducation\/2015\/09\/01\/the-secret-question-are-we-actually-good-at-math\/\"><span style=\"font-weight: 400\">I\u2019ve written about previously on this blog<\/span><\/a><span style=\"font-weight: 400\">. While the literature on mindset research <\/span><a href=\"http:\/\/psycnet.apa.org\/index.cfm?fa=buy.optionToBuy&amp;id=2012-20864-001\"><span style=\"font-weight: 400\">contains contradictory empirical findings<\/span><\/a><span style=\"font-weight: 400\">, this is not a problem for me since my main goal is to use the language and motivation that this research provides as a tool for engaging students across psychological domains. Mindset research is only one among many possible sources of motivation for meeting the goals above;<\/span><i><span style=\"font-weight: 400\"> what is critical is to make sure that my mathematics courses include activities that explicitly promote student development across all three of these psychological domains.<\/span><\/i><\/p>\n<p><b>A Toolbox of Interventions<\/b><\/p>\n<p><span style=\"font-weight: 400\">What follows are student assignments and activities that I\u2019ve used in classes ranging from 20-student upper-level courses for math majors to 150-student Calculus courses for STEM majors. They have a common purpose of promoting student development in one or both of the emotional or behavioral domains, complementing other work that my students do to develop intellectually in mathematics. An important disclaimer: none of these activities are original with me; rather, these are all adaptations of the work of others, to whom I will always be indebted.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Introductions<\/span><\/i><span style=\"font-weight: 400\">. On the first day of class each semester, I begin with students introducing themselves to each other. In a small class with less than 30-50 students, there is time for everyone to take turns sharing with the entire class their name and the reason they are taking the course. In a large-lecture course, I tell students to do the same thing with 4-6 people sitting next to each other. I teach at the University of Kentucky, and many of our STEM majors are primarily enrolled in large lecture courses during their first year. By beginning every course with a 5-minute activity that recognizes the students and promotes discussion, a collaborative tone is set for the remainder of the course, and some of the isolation that students feel (especially as one among many in a large lecture) can be countered.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Day 1, small classes: reading and autobiography assignment.<\/span><\/i><span style=\"font-weight: 400\"> During the first week of class, I assign an article regarding mindset research by Carol Dweck along with a one-page autobiographical essay. I have used Dweck\u2019s articles \u201c<\/span><a href=\"http:\/\/www.scientificamerican.com\/article\/the-secret-to-raising-smart-kids1\/\"><span style=\"font-weight: 400\">The Secret to Raising Smart Kids<\/span><\/a><span style=\"font-weight: 400\">\u201d and \u201c<\/span><a href=\"https:\/\/psychology.stanford.edu\/sites\/all\/files\/cdweckmathgift_0.pdf\"><span style=\"font-weight: 400\">Is Math a Gift? Beliefs that put females at risk<\/span><\/a><span style=\"font-weight: 400\">\u201d for this with success. I assign a grade to the essay based on completion only, completely ignoring the quality of the writing, editing, or ideas. The goal is to get students to reflect and be honest, not necessarily to train them to write well. If students respond to the prompt in a relevant manner, they get full credit.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Day 1, large classes: video and small group discussions.<\/span><\/i><span style=\"font-weight: 400\"> In large classes with 150 or more students, especially in courses that are coordinated across sections, the autobiography assignment is harder to implement. Another way to introduce students to the language of mindsets (or other tools) is to have students students watch a <\/span><a href=\"https:\/\/www.youtube.com\/watch?v=pN34FNbOKXc\"><span style=\"font-weight: 400\">10-minute video about mindset research<\/span><\/a><span style=\"font-weight: 400\"> during class on the first day. Following the video, have students spend 2-3 minutes free response writing about the video. Following the writing, have students spend 2-3 minutes discussing their response with a neighbor in the class. <\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Course policy on supportive language.<\/span><\/i><span style=\"font-weight: 400\"> I have a course policy on supportive language that I use in all of my classes: <\/span><i><span style=\"font-weight: 400\">Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason.<\/span><\/i><span style=\"font-weight: 400\"> I give students a variety of examples of \u201cbanned\u201d phrases and suggested replacements that can be found <\/span><a href=\"http:\/\/blogs.ams.org\/matheducation\/2015\/09\/01\/the-secret-question-are-we-actually-good-at-math\/\"><span style=\"font-weight: 400\">here<\/span><\/a><span style=\"font-weight: 400\">. The important aspect of this policy is that it must be enforced &#8212;\u00a0if I hear students making negative comments, I say \u201ccourse policy\u201d and have them create a neutral rephrasing of their negative self-comment. This is tougher to implement in large lectures, but even in this context the policy sets a positive tone for the first month of class. In large lectures with accompanying recitations, it is important that graduate student teaching assistants are aware of this policy and enforce it during their recitation sections. It is also important that students know that the policy applies to faculty and teaching assistants as well. I had a student in a large Calculus II lecture call me out for violating this policy last semester when I was frustrated at making errors during an example, and it was an excellent moment for the class.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Video regarding effectiveness of science videos.<\/span><\/i><span style=\"font-weight: 400\"> During class, I have students watch a <\/span><a href=\"https:\/\/www.youtube.com\/watch?v=eVtCO84MDj8\"><span style=\"font-weight: 400\">video about research regarding the effectiveness of science videos<\/span><\/a><span style=\"font-weight: 400\">. As with the video on the first day of class, students complete a two-minute free writing followed by a two-minute discussion with their neighbors regarding their response to the video. For many students, a common behavioral practice is that if they are stuck on a math problem, they immediately search the internet for videos that explain how to do this type of problem. This is typically an unproductive behavior, and dedicating some class time to confront it directly sets the stage for further discussions regarding the processes students use for completing homework and solving problems.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Assign an unsolved problem as homework.<\/span><\/i><span style=\"font-weight: 400\"> As <\/span><a href=\"http:\/\/blogs.ams.org\/matheducation\/2015\/05\/01\/famous-unsolved-math-problems-as-homework\/\"><span style=\"font-weight: 400\">I\u2019ve written before<\/span><\/a><span style=\"font-weight: 400\">, assigning an unsolved math problem as homework can serve as a gateway to discussions about the nature of high-level mathematical problem solving and the processes, practices, and attitudes that students bring to authentic mathematical challenges. When I assign an unsolved problem, e.g. those given in the article linked to above, I provide students with the following prompt.<\/span><\/p>\n<blockquote><p><i>This is a famous unsolved problem in mathematics. Work on it for a while \u2014 the goal isn\u2019t for you to solve this, but rather to get a feel for the problem. Create an essay by recording your thoughts and attempts as you work. Focus on responding to the following questions: What did you try to do? Why did you try this? What did you discover as a result? Why is this problem challenging? (Seriously, write down everything you\u2019re thinking and every idea you try, even if it doesn\u2019t go anywhere.)<\/i><\/p><\/blockquote>\n<p><span style=\"font-weight: 400\">It\u2019s good to grade this problem generously regarding mathematical content, keeping in mind that the goal is for students to be rewarded for demonstrating persistence and good mathematical processes.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Reflective essay about homework. <\/span><\/i><span style=\"font-weight: 400\">In most of my upper-level courses, especially those in which I assign an unsolved problem as homework, I have students write a 2-3-page essay explaining what they found most and least challenging in the homework so far, and what their most and least favorite homework problems have been. The prompt can ask them to directly link to mindset or another external topic, or can be left relatively open-ended to see what connections students make on their own. This can be either graded with a rubric for writing or graded based on completion. The majority of my students have discussed at length their experience working on the unsolved problem, both what they did and how they felt about their work.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Create-your-own homework assignment.<\/span><\/i><span style=\"font-weight: 400\"> A recent assignment that I\u2019ve used is to have students write their own homework assignment toward the end of the semester. The specific prompt I used was this:<\/span><\/p>\n<blockquote><p><i><span style=\"font-weight: 400\">Create your own homework assignment containing three problems. The homework assignment should be typed. There should be a mix of easy and hard problems that represent a broad spectrum of ideas from the entire course. For each of these problems, type a paragraph explaining why you chose that problem, whether you think it is easy, medium, or hard in difficulty, and what area of the course the problem represents. Once you have created the homework assignment, you should include complete solutions to each of the problems. Your solutions to the problems may be either typed or handwritten, but they should be complete and correct.<\/span><\/i><\/p><\/blockquote>\n<p><span style=\"font-weight: 400\">It was fascinating to see what the students came up with for their homework. What I found particularly noteworthy was the large number of students who included as one problem a critical analysis essay or short reflective essay similar to what I had assigned in the course to complement mathematical content work. I had honestly expected their assignments to contain a range of standard problems focused on mathematical content, and was pleasantly surprised to see the students incorporating into their homework tasks that addressed behavioral and emotional aspects of doing mathematics.<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">End-of-course reflective essay.<\/span><\/i><span style=\"font-weight: 400\"> In my smaller classes, I assign as the final homework assignment the following short essay prompt. The grade is based only on completion, because I want students to write honestly without fear of being penalized for their opinions.<\/span><\/p>\n<blockquote><p><i><span style=\"font-weight: 400\">What were six of the most important discoveries or realizations you made in this class? In other words, what are you taking away from this class that you think might stick with you over time and\/or influence you in the future? What have you experienced that might have a long-term effect on you intellectually or personally? These can include things you had not realized about mathematics or society, specific homework problems or theorems from the readings, etc. These can be things that made sense to you, or topics where you were confused, points that you agreed\/disagreed with in the readings or class discussions, issues that arose while working on your course project, etc. Explain why these six discoveries or realizations are important to you.<\/span><\/i><\/p><\/blockquote>\n<p><span style=\"font-weight: 400\">I have found that reading through these essays is a fascinating exercise, because of the wide range of messages that the students perceived as being central to the course. Using this assignment consistently over time has helped me improve my ability to create focused courses with clearly defined and communicated learning outcomes.<\/span><\/p>\n<p><b>Final Thought<\/b><\/p>\n<p><span style=\"font-weight: 400\">If you experiment with any of these activities in your own courses, I would love to hear about your experiences! <\/span><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>By Benjamin Braun, Editor-in-Chief, University of Kentucky In my experience, many students in K-12 and post-secondary mathematics courses believe that: all math problems have known answers, failure and misunderstanding are absent from successful mathematics, their instructor can always find answers &hellip; <a href=\"https:\/\/blogs.ams.org\/matheducation\/2016\/05\/16\/believing-in-mathematics\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/matheducation\/2016\/05\/16\/believing-in-mathematics\/><\/div>\n","protected":false},"author":73,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[28,27],"tags":[23,216],"class_list":["post-1261","post","type-post","status-publish","format-standard","hentry","category-assessment-practices","category-classroom-practices","tag-active-learning","tag-psychology"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6C2AC-kl","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts\/1261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/users\/73"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/comments?post=1261"}],"version-history":[{"count":8,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts\/1261\/revisions"}],"predecessor-version":[{"id":1270,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/posts\/1261\/revisions\/1270"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/media?parent=1261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/categories?post=1261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/matheducation\/wp-json\/wp\/v2\/tags?post=1261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}